라플라스 변환 표 (Laplace Transform Table)

:: Laplace Transform Table ::

Function	Laplace Transform	Remarks
$$f(x)$$	$$\mathcal{L}\{f\}(s) = \int_0^\infty f(x)e^{-sx}dx$$	Definition
$$af(x) +bg(x)$$	$$a\mathcal{L}\{f\}(s) + b\mathcal\{g\}(s)$$	Linearity
$$xf(x)$$	$$-\left(\mathcal{L}\{f\}\right)'(s)$$
$$x^nf(x)$$	$$(-1)^n\left(\mathcal{L}\{f\}\right)^{(n)}(s)$$
$$f'(x)$$	$$s\mathcal{L}\{f\}(s) - f(0)$$
$$f^{(n)}(x)$$	$$s^n\mathcal{L}\{f\}(s) - \sum_{k=1}^ns^{n-k}f^{(k-1)}(0)$$
$$\frac{f(x)}{x}$$	$$\int_s^\infty \mathcal{L}\{f\}(\sigma)d\sigma$$
$$\int_0^xf(t)dt$$	$$\frac{\mathcal{L}\{f\}(s)}{s}$$
$$e^{ax}f(x)$$	$$\mathcal{L}\{f\}(s-a)$$
$$f(x-a)u(x-a)$$	$$e^{-as}\mathcal{L}\{f\}(s)$$
$$f(ax)$$	$$\frac{1}{a}\mathcal{L}\{f\}(s)$$
$$(f*g)(x)$$	$$\mathcal{L}\{f\}(s)\mathcal{L}\{g\}(s)$$

Function	Laplace Transform	Remarks
$$f(x)$$	$$\mathcal{L}\{f\}(s) = \int_0^\infty f(x)e^{-sx}dx$$	Definition
$$\delta(x)$$	$$1$$	$\delta$ : Dirac Delta Function
$$u(x)$$	$$\frac{1}{s}$$	$u(x)$ : Step Function
$$\sqrt[n]{x}u(x)$$	$$\frac{1}{s^{\frac{1}{n}+1}}\Gamma\left(\frac{1}{n}+1\right)$$
$$\sin(a x)u(x)$$	$$\frac{a}{s^2+a^2}$$
$$\cos(ax)u(x)$$	$$\frac{s}{s^2+a^2}$$
$$\sinh(ax)u(x)$$	$$\frac{a}{s^2-a^2}$$
$$\cosh(ax)u(x)$$	$$\frac{s}{s^2-a^2}$$

Wikipedia의 Laplace Transform 항목을 참고하였습니다.